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Hardware-efficient variational quantum algorithms for time evolution

Marcello Benedetti, Mattia Fiorentini, Michael Lubasch

arXiv:2009.12361v5quant-ph

TL;DR

Variational time-evolution methods based on the time-dependent variational principle require matrix inversion, which can be challenging for ill-conditioned matrices. This paper presents alternatives that avoid inversion and adjust hardware requirements to accuracy needs; its numerical results show strong imaginary-time performance and high-accuracy real-time updates.

  • Problem

    Existing time-dependent variational-principle approaches to variational time evolution require matrix inversion, which poses computational challenges for ill-conditioned matrices.

    Source: §I · p. 1

  • Method

    The paper develops alternatives that sequentially optimize Trotter terms using causal cones and coordinatewise updates, without requiring matrix inversion.

    Source: §I · pp. 1–2

  • Results

    Angle update performs remarkably well for imaginary-time ground-state search, while block and cone updates achieve high real-time accuracy; the illustrated imaginary-time run has relative energy errors below 10^-3.

    Source: §I · p. 1, §IV · p. 7

  • Takeaways & Limitations

    Causal cones enable finite-system simulations larger than the available quantum hardware, and algorithm hardware requirements can be adjusted to experimental capabilities.

    Source: §IV · pp. 7–8

Abstract

from arXiv · show

Parameterized quantum circuits are a promising technology for achieving a quantum advantage. An important application is the variational simulation of time evolution of quantum systems. To make the most of quantum hardware, variational algorithms need to be as hardware-efficient as possible. Here we present alternatives to the time-dependent variational principle that are hardware-efficient and do not require matrix inversion. In relation to imaginary time evolution, our approach significantly reduces the hardware requirements. With regards to real time evolution, where high precision can be important, we present algorithms of systematically increasing accuracy and hardware requirements. We numerically analyze the performance of our algorithms using quantum Hamiltonians with local interactions.

I. INTRODUCTION

The paper targets hardware-efficient variational time evolution, replacing matrix-inversion-based approaches with alternatives whose accuracy and hardware requirements can be adjusted. It draws on sequential Trotter-term optimization, causal cones, and coordinatewise updates, and reports different strategies for imaginary and real time evolution.

I. INTRODUCTION

  • Time-evolution simulation is motivated by its potential for quantum advantage and its relevance to quantum chemistry and materials science.The paper frames reducing experimental requirements without sacrificing accuracy as a central design challenge.Source: §I · p. 1
  • The proposed alternatives avoid matrix inversion and let hardware requirements increase systematically alongside simulation accuracy.The paper identifies ill-conditioned matrices as a computational challenge for existing time-dependent variational-principle approaches.Source: §I · pp. 1–2
  • With depth D=2, the illustrated imaginary-time simulation reaches relative energy errors below 10^-3, and the real-time simulation ends at squared Euclidean distance 0.05.The examples use causal cones requiring six qubits, for chain lengths n=8, 10, and 12.Source: §I · p. 1, Figure 1 · p. 1
  • The optimization strategy processes Trotter terms sequentially, restricts each optimization to its causal cone, and updates parameter groups coordinatewise.The paper relates these choices to tensor-network methods including time-evolving block decimation and matrix product states.Source: §I · p. 2
  • Here, “hardware-efficient” includes device-tailored circuits or techniques such as causal cones that reduce qubit or gate requirements.The introduction also mentions measurement-and-reset schemes and methods for periodic systems as alternative efficient approaches.Source: §I · p. 2

A. Taking apart time evolution

The method approximates time evolution by applying short-time Trotter terms sequentially and variationally fitting the state after each term. It repeats this procedure across all terms and time steps.

A. Taking apart time evolution

  • Imaginary-time evolution is obtained by substituting t with −it in the real-time formulation.Source: A. Taking apart time evolution · p. 2
  • The full evolution is approximated by repeated short-time products U(τ), with τ=t/N, and the Trotter approximation can be made more accurate using higher-order formulas.Each product orders exponentials of the Hamiltonian terms.Source: A. Taking apart time evolution · p. 2
  • For each Trotter term, the method varies θ to minimize the squared distance between the variational state and the term applied to the previous variational state.The resulting optimization is expressed as maximizing an objective function.Source: A. Taking apart time evolution · p. 2
  • The procedure optimizes all K terms in each of N time steps, completing the evolution after the final step.Figure 2 depicts successive Trotter terms acting on the initial variational state, with parameters optimized at each step.Source: A. Taking apart time evolution · p. 2

B. Parameterized quantum circuits and causal cones

The variational states use nearest-neighbor two-qubit blocks, while causal cones restrict local expectation calculations and updates to a smaller circuit region. This can reduce the hardware footprint below the size of the simulated system.

B. Parameterized quantum circuits and causal cones

  • The ansatz uses generic two-qubit unitaries on nearest neighbors, matching one-dimensional connectivity found on many quantum computers.Increasing circuit depth makes these ansätze universal for quantum computation.Source: B. Parameterized quantum circuits and… · p. 2
  • A local operator’s expectation depends only on blocks inside its causal cone, allowing a two-qubit expectation to be estimated with six qubits and five blocks regardless of ansatz size.Source: B. Parameterized quantum circuits and… · p. 2
  • Optimizing only blocks inside each Trotter term’s causal cone restricts the required circuit size and permits variational states larger than the available quantum hardware.Causal cones can also accommodate periodic boundary conditions without additional hardware requirements; required physical qubits depend on ansatz depth and the Trotter term.Source: B. Parameterized quantum circuits and… · pp. 2–3
  • The considered ansatz starts at depth D=2, and its depth can be increased by appending blocks initialized to 1 when variational error grows.This allows the ansatz to adapt its expressive power during time evolution.Source: B. Parameterized quantum circuits and… · p. 3

C. The coordinatewise update rule

The coordinatewise objective is sinusoidal in each parameter when the others are fixed, so its local maximum can be found without gradients or Hessians. The update rule exploits the previous objective value to reduce the search to one evaluation.

C. The coordinatewise update rule

  • The standard parameterized-gate form makes the objective sinusoidal when one parameter varies, giving an analytic expression for its extrema.This property had previously been shown for energy expectation values.Source: C. The coordinatewise update rule · p. 3
  • With all other parameters fixed, the objective is sinusoidal in the selected parameter, enabling coordinatewise optimization without gradients or Hessians.Each sweep sets parameters to their locally optimal values.Source: C. The coordinatewise update rule · p. 3
  • The update rule estimates the sinusoid's amplitude and phase, then uses the known current objective value to find the maximum with a single evaluation.The direct formula initially calls for evaluations at the current parameter and at a shift of π/2; the previous-step value supplies the former.Source: C. The coordinatewise update rule · p. 4

D. Hardware-efficient implementation

The Hadamard test estimates the objective and update rule, but is simpler when consecutive circuits differ only locally. Cone, block, and angle updates progressively reduce their differences, with angle update permitting an ancilla-free direct-measurement variant.

D. Hardware-efficient implementation

  • The Hadamard test estimates the objective and update rule, but unrelated circuits can require many controlled operations.Local transformations between circuits simplify the test.Source: D. Hardware-efficient implementation · p. 4
  • Cone update optimizes only blocks in the Trotter term's causal cone; its Hadamard test requires O(NbNp) controlled gates.Nb is the number of blocks in the cone and Np the parameters per block.Source: D. Hardware-efficient implementation · p. 4
  • Block update replaces the previous state with the current state after each block update, leaving circuits differing by O(Np) parameters and using τ/Nb in the objective.The division is unnecessary when τ is a hyperparameter, as in imaginary time evolution.Source: D. Hardware-efficient implementation · p. 4
  • Angle update replaces the previous state after each parameter update, leaving circuits differing by one parameter and using τ/(NbNp) in the objective.When τ is a hyperparameter, division is unnecessary; direct measurements can also remove the ancilla and controlled gate.Source: D. Hardware-efficient implementation · p. 4
  • Table I compares cone, block, and angle updates with TDVP methods; a sweep updates every parameter in a Trotter term's causal cone once.Source: Table I · p. 5

A. Error analysis

The method has Trotter error from splitting the evolution operator and variational error from the ansatz and optimization. Numerical evidence shows second-order Trotter evolution gives smaller errors than first-order across tested time steps.

A. Error analysis

  • The method has Trotter error from splitting the evolution operator and variational error from limitations of the ansatz and optimization.Both errors can be quantified and systematically decreased.Source: A. Error analysis · p. 5
  • For a pth-order Trotter formula, total Trotter error scales as O(τ^p) over N = t/τ steps when τ is sufficiently small.The per-step scaling is O(τ^(p+1)); higher-order formulas can reduce Trotter error.Source: A. Error analysis · p. 5
  • Variational error depends on ansatz expressiveness and can be reduced by adding PQC blocks or performing more optimization sweeps.A sweep updates every parameter inside the Trotter term's causal cone once.Source: A. Error analysis · p. 5
  • Second-order Trotter evolution produced smaller squared-distance errors than first-order for every tested time step.The test used real-time evolution of a six-qubit quantum Ising chain to t = 2, with cone update and six sweeps.Source: A. Error analysis · p. 5
  • Machine-learning heuristics have also been used to assist time-evolution algorithms and reduce variational and Trotter errors.Source: A. Error analysis · p. 5

B. Comparison of hardware requirements

Angle update uses the fewest resources per sweep among the proposed updates, and the updates avoid matrix inversion required by TDVP methods. The comparison also highlights the measurement burden of ill-conditioned TDVP matrices and the trade-off between Trotter and accumulated variational error.

B. Comparison of hardware requirements

  • At large τ, Trotter error exceeds variational error; at small τ, accumulated variational error rapidly becomes larger.The comparison uses real-time evolution to t = 2 with first- and second-order Trotter formulas.Source: Figure 5 · p. 5
  • Angle update requires the smallest per-sweep resources among the proposed methods and no matrix inversion; block update interpolates between angle and cone update.Source: B. Comparison of hardware requirements · p. 5
  • The illustrated Trotter-term positions have causal cones of four qubits and three blocks (45 parameters), or six qubits and five blocks (75 parameters).Source: B. Comparison of hardware requirements · p. 6
  • For ill-conditioned TDVP matrices, achieving parameter accuracy ϵ can require O(κ^2/ϵ^2) measurements, where κ is the matrix condition number.Matrix inversion can magnify small errors in the matrix elements.Source: B. Comparison of hardware requirements · pp. 5–6
  • The median TDVP condition number rises from about 16 with 15 parameters to about 5,001 with 45 and 17,085 with 75.These estimates use 100 random initializations and ignore singular values below 10^-7.Source: B. Comparison of hardware requirements · p. 6

C. Numerical experiments

Cone updates attain the optimal objective for four-qubit cones, while six-qubit cones remain limited by the shallow Ansatz. In imaginary-time tests, all update methods reach low energies, and angle-update circuits are simpler than cone-update circuits.

C. Numerical experiments

  • For four-qubit cones, cone update converges to the optimal objective of one; for six-qubit cones, no method reaches one, reflecting the shallow Ansatz’s limitations.Block and angle updates do not benefit from more sweeps in the four-qubit case.Source: C. Numerical experiments · p. 7
  • At the critical point, all methods reach similar low energies; away from criticality, all converge rapidly to states very close to the ground state.The tests use J = λ = 1 at the critical point and J = 1, λ = 4 away from it.Source: C. Numerical experiments · p. 7
  • Angle-update circuits are much simpler than cone-update circuits.Source: C. Numerical experiments · p. 7

IV. DISCUSSION

The discussion presents matrix-inversion-free algorithms with hardware demands that can be adjusted to experimental capabilities. The appendices describe sequential Trotter-term variational updates and a sinusoidal coordinatewise objective whose maxima can be found in closed form.

IV. DISCUSSION

  • The algorithms avoid inverting a possibly ill-conditioned matrix at each time step, and their hardware requirements can be adjusted to experimental capabilities.Source: §IV · p. 7
  • For imaginary-time ground-state search, angle update performs remarkably well; after convergence, block or cone updates can fine-tune the result.Source: §IV · p. 7
  • For real-time Schrödinger evolution, numerical evidence supports high accuracy from block and cone updates, while angle update may suit applications where per-step accuracy is not crucial.Steady-state computation is given as an example of such an application.Source: §IV · pp. 7–8
  • The work brings PQC and tensor-network optimization closer together, drawing on tensor-network ideas including Trotter-term sequencing and causal cones.Source: §IV · p. 8
  • Causal cones are used to simulate finite systems larger than the underlying quantum hardware, and are also an ingredient in noise-resilient circuit construction [34] [63].Source: §IV · p. 8

Appendix A: Variational simulation of time evolution

  • Appendix A formulates evolution for arbitrary complex time, then specializes to real and imaginary time and approximates the Trotter product by optimizing one operation after another.Source: Appendix A · p. 8
  • The variational step minimizes squared Euclidean distance, equivalent to maximizing an objective, assuming constant Ansatz norm and Hermitian Hamiltonian terms.Source: Appendix A · pp. 8–9

Appendix B: Sinusoidal form of the coordinatewise objective function

  • For coordinatewise optimization, the objective is sinusoidal in one parameter, so its maxima are closed-form and require evaluations at two parameter values spaced π apart.Source: Appendix B · p. 10, Figure 9 · p. 10

Appendix C: The update rule with a single evaluation

These appendices describe coordinatewise update rules and measurement circuits for variational time evolution, then derive the TDVP linear system and specify the numerical simulation setup.

Appendix C: The update rule with a single evaluation

  • Reusing the preceding coordinate update lets the single-evaluation rule optimize each parameter using information from the previous step.The coordinatewise objective is sinusoidal, and the update rule is stated for variational simulation of time evolution.Source: Appendix C · pp. 10–11

Appendix D: Hadamard test for cone update

  • Cone update exploits the local difference between successive circuits to simplify the Hadamard test, which uses O(NbNp) controlled operations for a cone with Nb blocks of Np parameterized gates.The test estimates the real and imaginary parts of the relevant overlap using two circuits.Source: Appendix D · p. 11

Appendix E: Hadamard test for angle update

  • Angle update replaces ancilla qubits and controlled gates with direct measurements, at the cost of executing additional circuits.For imaginary time, approximate recycling reduces the circuit count to two; its error is proportional to the previous parameter change and can be reduced by decreasing τ.Source: Appendix E · pp. 11–12

Appendix F: Time-dependent variational principle and matrix inversion

  • TDVP expresses parameter time evolution as a linear system, and matrix inversion makes the measurement cost scale as O(κ²/ϵ²) for accuracy ϵ.The extra κ² factor relative to O(1/ϵ²) measurement error can be challenging for ill-conditioned A.Source: Appendix F · pp. 13–14
  • Finite measurements make TDVP’s matrix A and vector C estimates imprecise, with mean-value sampling error scaling as O(1/√m).The appendix analyzes how errors in these measured quantities affect the solution vector B.Source: Appendix F · pp. 13–14

Appendix G: Numerical simulations

  • The simulations use depth-2 circuits whose causal cones involve at most six qubits, and omit both finite-sampling and hardware noise.Expectations and Hadamard tests are calculated exactly.Source: Appendix G · pp. 14–15
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